Nuprl Lemma : trivial-Taylor-approx
∀[I:Interval]. ∀[n:ℕ]. ∀[F:ℕn + 1 ⟶ I ⟶ℝ]. ∀[a,b:{x:ℝ| x ∈ I} ].
((a = b) ⇒ (Taylor-approx(n;a;b;i,x.F[i;x]) = F[0;a]))
Proof
Definitions occuring in Statement :
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
Taylor-approx: Taylor-approx(n;a;b;i,x.F[i; x]),
prop: ℙ,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
nat: ℕ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
fact: (n)!,
primrec: primrec(n;b;c),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
subtract: n - m,
rsub: x - y
Lemmas referenced :
req_wf,
req_witness,
Taylor-approx_wf,
int_seg_wf,
rfun_wf,
i-member_wf,
real_wf,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
set_wf,
nat_wf,
interval_wf,
rsum_wf,
rmul_wf,
rdiv_wf,
int-to-real_wf,
fact_wf,
int_seg_subtype_nat,
rless-int,
int_seg_properties,
le_wf,
nat_plus_properties,
rless_wf,
rnexp_wf,
rsub_wf,
radd_wf,
nat_plus_wf,
decidable__le,
rneq-int,
fact-non-zero,
fact0_redex_lemma,
rnexp_zero_lemma,
req_weakening,
req_functionality,
rsum-split-first,
radd_functionality,
req_transitivity,
rsum-zero,
uiff_transitivity,
rmul-rdiv-cancel2,
radd_comm,
radd-zero-both,
rsum_functionality,
equal_wf,
rmul_functionality,
rnexp_functionality,
rsub_functionality,
rminus_wf,
exp_wf2,
not-lt-2,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
zero-add,
add_functionality_wrt_le,
le-add-cancel,
less_than_wf,
rmul-zero-both,
rmul_comm,
radd-rminus-both,
rnexp-int,
squash_wf,
true_wf,
exp-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
addEquality,
because_Cache,
dependent_set_memberEquality,
setEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
inrFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
minusEquality,
imageElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n + 1 {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}]. \mforall{}[a,b:\{x:\mBbbR{}| x \mmember{} I\} ].
((a = b) {}\mRightarrow{} (Taylor-approx(n;a;b;i,x.F[i;x]) = F[0;a]))
Date html generated:
2017_10_03-PM-00_38_09
Last ObjectModification:
2017_07_28-AM-08_45_03
Theory : reals
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